We propose a one-parameter family \ $\mathbb{R}_q$ \ of deformations of the
reals, which is motivated by the generalized additivity of the Tsallis entropy.
We introduce a generalized multiplication which is distributive with respect to
the generalized addition of the Tsallis entropy. These operations establish a
one-parameter family of field isomorphisms \ $\tau_q$ \ between \ $\mathbb{R}$
\ and \ $\mathbb{R}_q$ \ through which an absolute value on \ $\mathbb{R}_q$ \
is introduced. This turns out to be a quasisymmetric map, whose metric and
measure-theoretical implications are pointed out.Comment: 16 pages, Standard LaTeX2e, To be published in Physica
We construct generalized additions and multiplications, forming fields, and division algebras inspired by the Tsallis thermo-statistics. We also construct derivations and integrations in this spirit. These operations do not reduce to the naively expected ones, when the deformation parameter approaches zero.
We present a geometric argument that explains why some systems having
vanishing largest Lyapunov exponent have underlying dynamics aspects of which
can be effectively described by the Tsallis entropy. We rely on a comparison of
the generalised additivity of the Tsallis entropy versus the ordinary
additivity of the BGS entropy. We translate this comparison in metric terms by
using an effective hyperbolic metric on the configuration/phase space for the
Tsallis entropy versus the Euclidean one in the case of the BGS entropy.
Solving the Jacobi equation for such hyperbolic metrics effectively sets the
largest Lyapunov exponent computed with respect to the corresponding Euclidean
metric to zero. This conclusion is in agreement with all currently known
results about systems that have a simple asymptotic behaviour and are described
by the Tsallis entropy.Comment: 15 pages, No figures. LaTex2e. Some overlap with arXiv:1104.4869
Additional references and clarifications in this version. To be published in
QScience Connec
Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N -Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a connection with the non-additive entropy under investigation. We present an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor.
We present an embedding of the Tsallis entropy into the 3-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.
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